Homework 11

(to be turned in on May 6 as part of WPR III Grade)

Sequences of Functions and Series

1. Consider continuous functions from $[0,1]$ to $\mathbb{R}$ satisfying $0\leq f_1\leq f_2\leq f_3 \leq \cdots$ which converge pointwise to f. Must f be continuous? Give a proof or a counterexample.

2. Prove convergence and find the limit, or prove that the series does not converge:

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3. What are the possible values of rearrangements of the following series?

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The Lipschitz Condition

A function is said to be Lipschitz with Lipschitz constant C if $|f(x)-f(y)|\leq C|x-y|$ for all x,y in the domain. (See Wikipedia article on Rudolf_Lipschitz.)

4. What does the Lipschitz condition say about the secant lines of f?

5. Prove that if f is Lipschitz on a particular domain, then f is uniformly continuous on that domain.

6. Prove that if the derivative of a function f exists and is bounded on some domain, that is $|f'(x)|\leq M$ for some M, then f is Lipschitz on that domain. How does the Lipschitz constant relate to the bound M?

7. Does the reverse hold true? That is, if f is Lipschitz, must the derivative (provided it exists) be bounded? Prove or give a counterexample.

8. Let $(f_n)$ be a pointwise convergent sequence of Lipschitz functions on $[0,1]$ with Lipschitz constant C. Prove that the sequence converges uniformly. Will the limit $\lim_{n\to\infty} f_n$ also be Lipschitz?

9. Let $x,y$ be fixed. Suppose f is Lipschitz with constant C. Define

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What can you say about the limits of $f_n(x)$ and $f_n(y)$ for $n\to\infty$ if (a) $C<1$ or (b) $C\geq 1$?